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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习 4.1 - 可测函数及其性质 }
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\date{2024 年 4 月 29 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\vspace{-0.5cm}

\begin{enumerate}

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\item  %Problem 01
实变函数研究的函数的定义域和值域分别是什么？
设 $f$ 是定义在可测集 $E\subseteq\mathbb{R}^n$ 的实函数，什么时候称 $f$ 是可测函数？

\vspace{0.05cm}

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\item  %Problem 02
设 $f$ 是定义在可测集 $E\subseteq\mathbb{R}^n$ 的实函数。证明下述条件相互等价： 
\begin{enumerate}
\item  $f$ 是可测函数。 
\item  对任意实数 $a$, $E[f\ge a]$ 都是可测集。 
\item  对任意实数 $a$, $E[f<a]$ 都是可测集。 
\item  对任意实数 $a$, $E[f\le a]$ 都是可测集。 
\item  对任意实数 $a,b, (a<b)$, 集合 $E[a\le f<b]$ 都是可测集。 
\end{enumerate}

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\item  %Problem 03
设 $f(x)$ 是可测集 $E$ 上的可测函数，设 $a$ 是任意实数或 $\pm\infty$. 证明 $E[f=a]$ 是可测集。

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\item  %Problem 04
证明区间 $[a,b]\subseteq\mathbb{R}$ 上的连续函数是可测函数。

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\item  %Problem 05
什么是可测集 $E\subseteq\mathbb{R}^n$ 上的连续函数？证明可测集 $E$ 上的连续函数是可测函数。

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\item  %Problem 06
设 $f(x)$ 是可测集 $E$ 上的可测函数，设 $F\subseteq E$ 是可测子集。证明限制 $f\mid_F$ 也是可测函数。

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\item  %Problem 07
设 $f(x)$ 定义在有限个可测集 $E_i (1\le i\le s)$ 的并集 $E$ 上，并且 $f(x)$ 看作定义在每个 $E_i$ 上的函数时都是可测的。证明 $f(x)$ 看作定义在 $E$ 上的函数时也是可测的。

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\item  %Problem 08
设 $f$ 是定义在可测集 $E\subseteq\mathbb{R}^n$ 的实函数。什么时候称 $f(x)$ 是简单函数？

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\item  %Problem 9
设 $f(x)$ 与 $g(x)$ 是可测集 $E$ 上的可测函数。证明 $E[f>g]$ 与 $E[f\ge g]$ 都是可测集。

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\item  %Problem 10
设 $f(x)$ 与 $g(x)$ 是可测集 $E$ 上的可测函数。证明 $f(x)+g(x)$ 与 $f(x)g(x)$ 也是 $E$ 上的可测函数。

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\item  %Problem 11
设 $\{f_n(x)\}$ 是可测集 $E$ 上的一列可测函数。证明  
$\underset{n}{\inf} f_n(x)$ 与 $\underset{n}{\sup} f_n(x)$ 也是 $E$ 上的可测函数。

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\item  %Problem 12
设 $\{f_n(x)\}$ 是可测集 $E$ 上的一列可测函数。证明 $\varliminf\limits_{n\to\infty} f_n(x) $ 与 $\varlimsup\limits_{n\to\infty} f_n(x)$ 也是 $E$ 上的可测函数。

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\item  %Problem 13 定理7a（可测函数与简单函数的关系）
证明： 设 $f(x)$ 是可测集 $E$ 上的非负可测函数，则存在可测的简单函数序列 $\{\varphi_k(x)\}$, 使得对任意 $x\in E$, 
有 $\varphi_1(x)\le \varphi_2(x)\le \cdots\le \varphi_k(x)\le \cdots$, 且 $\lim\limits_{k\to\infty} \varphi_k(x)=f(x).$    

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\item  %Problem 14 定理7b（可测函数与简单函数的关系）
证明： 设 $f(x)$ 是可测集 $E$ 上的可测函数，则存在可测的简单函数序列 $\{\varphi_k(x)\}$, 使得对任意 $x\in E$, 
有 $\lim\limits_{k\to\infty} \varphi_k(x)=f(x)$. 若 $f(x)$ 还在 $E$ 上有界，则上述收敛可以是一致的。


\vspace{0.05cm}

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\item  %Problem 15
判断下述命题是否正确： 
\begin{enumerate}
\item   $|\tan(x)|<\infty$ 在 $E=\mathbb{R}$ 上几乎处处成立。  
\item   定义在区间 $E=[0,1]$ 上的狄利克雷函数 $D(x)$ 几乎处处等于零。  
\end{enumerate}

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\item  %Problem 16 习题1
证明 $f(x)$ 在可测集 $E$ 上是可测函数的充分必要条件是对任意有理数 $r$, 集合 $E[f>r]$ 是可测集。
如果对任意有理数 $r$, 集合 $E[f=r]$ 可测，问函数 $f(x)$ 是否可测？

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\item  %Problem 17 习题3
设 $E$ 是 $[0,1]$ 中的不可测集。设 $f(x)=\left\{\begin{array}{ll} x, & x\in E, \\ -x, & x\in [0,1]-E, \end{array}\right. $ 问 $f(x)$ 在 $[0,1]$ 上是否可测？问 $|f(x)|$ 是否可测？

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\item  %Problem 18 习题6
设 $f(x)$ 在 $(-\infty,\infty)$ 上连续，设 $g(x)$ 在可测集 $E\subseteq\mathbb{R}^n$ 上有限可测，则 $f(g(x))$ 在 $E$ 上可测。

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\end{enumerate}


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\end{document}

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